The Lorentz invariant measure and the Heisenberg uncertainty principle BY PATRICK BRUSKIEWICH AND ALICIA KIEtBIK Abstract A novel approach to the Heisenberg uncer- tainty principle is presented in this paper in a fashion that directly links special relativity and quantum theory from first principles. A direct link between special relativity and quantum theory The traditional approach to non-relativistic quan- tum mechanics is primarily didactic. The basic axioms of the theory are introduced, along with the Heisenberg uncertainty principle and simple systems such as the quantum harmonic oscilla- tor, are elucidated. A different approach, which is heuristic in nature, is to begin with the Lorentz invariant mea- sure from special relativity. The Lorentz invariant measure can be used as a conceptual foundation of quantum mechanics, and a direct link between special relativity and quantum, mechanics can be presented without recourse to subsidiary con- ditions. As a result of this heuristic approach, a foundation can be laid that firmly joins the two fields, which are typically presented separately, until second quantization and the Klein-Gordon and Dirac equations are presented. The Lorentz invariant measure As outlined, in this paper, it appears evident thai special relativity and quantum mechanics can be directly linked from first principles. Starting with the Lorentz invariant measure, a novel approach to the Heisenberg uncertainty principle is presented. The Lorentz invariant measure is 1 2 2 2 2 / 2 1 2 \ ex =c r -lx +y +z \, (1) which joins two points in Minkowski space. In this paper, we can look at the xt-plane without any loss of generality. For a one-dimensional inertial system, 2 2 2^2 2 f-)\ C T =C t~ -x . U) Multiplying through by mass m, and rear- ranging terms, we get 2 2.2 22 / n \ mx -mc i = -mc x . (3) Take the partial derivative of this expression with respect to X, the proper time ("temps proper" which literally means one's own time) and sim- pl i fy. This yields dx 2 dt xm tmc — = —mc " x. dx dx 14) From special relativity, m(dx/dx) = my(3x/9t) = ymv = p, and (dt/dx) = y, so then xp ~Et = -mc~% (5) is also invariant. In 3D, this is the familiar inner product of the two four- vectors, namely x Pm p =-mc 1 x. (6) In quantum mechanics, this measure is impor- tant when describing a plane wave (unction \|/ =exp i(px—Et) ft = exp i*-p>) (7! or a wave packet made up of a superposition of states such as j/K/?)exp i^p») d N p, (8) w T here A(p) is a momentum dependent amplitude, and N is the dimension of the space. By inspection, it is evident, then, that x P, (9) represents a phase (p. Measurement in quantum mechanics involves both amplitude and phase considerations. For information to be pass between two events in space-time a signal must be exchanged between the two events. The fastest signal that can be exchanged travels at the speed of light. As a signal, or wave, travels between two points in space, its change in phase may be used as a measure of the separation between the two points in space-time. Phase difference between adjacent events in Minkowski space-time In studying quantum measures, we can use dif- ferent approaches, such as the limiting process and infinitesimal analysis, a technique developed at Cambridge beginning in Newton's time. In this paper, we shall use. infinitesimal analysis. 2 In recent times, some of the infinitesimal analy- sis techniques have also been used in Wiener measures and Ito calculus in stochastic quantum mechanics. In ID, the difference in phase between two adjacent events in Minkowski space-time, two points infinitesimally separated by a causal connection (i.e. two points on or within the light cone connecting the two events) is 3 5(p « <f> 2 -<p, = (x 2 p 2 - E 2 t 2 ) - (x, p } - E,t\ ) = (x l +§x)(p l +dp)-{E l +bE)(t l +8t) = bxp { +xfip+ 8x8p - 3Et } - E { 8t - 8E8t. (10) This expression can be grouped into three terms First, consi der the bEt } term. Given thai E - • N /pV+ m V , then 5£ - 1 5p = v 5p - _J- qV, E t (ID where we have used p = ymv and E = ymc 2 . Since v = 5x/5t, this means vt, = x } . So for two points infinitesimally separated by a causal connection, X[8p - 5Et[ - and these two terms cancel each other out. in the expression. Consider the E,or term: ( -< 2 ^ £,5/ = ymc 2 3t = (ymv ) — 6/ = pfix, \ i J (12) where we have used the group velocity v g given by v = c 2 /v g . For a wave disturbance, whether it is a massless particle like a photon or a massive par- ticle like an electron, it is the group velocity that characterizes the momentum or energy o{ the w T ave packet, and therefore the exchange of infor- mation between two points in space-time. For two points infinitesimally separated by a causal connection, Efit - p { 3x = and these two terms cancel each other out in the expression. Of the six terms in the infinitesimal phase expression, only two terms remain: the differ- ence in phase between two points infinitesimally separated by a causal connection in space-time given by 8(p~5x6>-8£5/. (13) From the standpoint of infinitesimals, this expression is counter-intuitive in that the phase is made up of a product of terms and we would expect an infinitesimal change in the phase to be first order in these terms. The fact the first-order 16 Canadian Undergraduate Physics Journal Volume VI Issue 3 APRlt2008 infinitesimal terms drop out of the expression is a direct result of the Lorentz invariant mea- sure and appears to form the conceptual basis to the Heisenberg uncertainty principle. It is at this juncture that special relativity and quantum mechanics are linked at first principles. For a massless particle like a photon, then 8E - cSp => 8Ebt = Spcdt = dpdx. (14) As is evident, near an extremum where Sep is small, then dEdt « 8p8x. (15) Based on our derivation from the invariant measure, there is nothing that predisposes that Sep = 0, 8E8t ~ 0, or 8p8x ~ 0. When we consider that information exchange requires the exchange of energy or momentum, then by dimensional analysis, we find 8E8t « 5>Sx ~ h. (16) The classical standpoint Let us consider this expression from a classical standpoint, in terms of generalized momentum p and generalized q. By the equivalence of work- energy, we have \dE=\Fdq=\^d q . (17) We consider the phase space of a quasi-sta- tionary or periodic system and take a time aver- age {dp . J**-* J**- (18) cance of this remarkable constant." 1 The Dfrac delta potential It is straightforward to use this quantum action to solve for the energy states of a particle in a box or for a quantum harmonic oscillator. A more chal- lenging system to studying using the phase space approach is that of a particle trapped by a Dirac delta potential. Consider a potential of finite size and shrink the region of interaction to a point. Expressed as a ID attractive Dirac delta potential of strength a the Schrodinger equation for this system is ft 2 d 2 - \|/-a5(x)y = £\|/. (21) 2m ax There are a number of ways to solve this second order di fferential equation, including using integral transforms. The integral transform methods are outlined in a paper titled "A Novel Look at the One Dimensional Delta Schrodinger Equation/' written by one of the authors for the Gamma Magazine of the Niels Bohr Institute in May 2005. 5 As is well known, this equation has the single solution i|/(x) = \|/(0)exp By inspection, px ma\x\ -ma x ma (22) (23) ft ft 2 n and so we can solve for the energy of this state, namely N- di J 5/ This implies then that 5/ jdE » jdpdq ^> StdE w dpdq. (19) The classical concept of phase space is consis- tent with the derivative from the Lorentz invari- ant measure. At the Solvay Conference in 1911, Max Planck introduced the quantum action, namely jdpdq = nth (20) where n is an integer and ti is Planck's constant. This phase integral represents the quantization of quantum action. Planck expressed the view that "one should therefore confine oneself to the prin- ciple that the elementary region of probability, ft 8p8x has an ascertainable finite value and avoid any further speculation about the physical signifi- WWW.CUPJ.CA {ma I h) tmx^ (24) 2m 2m 2ft 2 The velocity of the particle can be derived (25) from ma a 8x p = mv = => v - — = — . ti ft 8t 'fake the derivative of the momentum with respect to the potential strength a: 8am 8p = - (26) if we multiply both sides of this equation with ox = abl/ft, then _( 8am Va5/ "I » A » (27) \ We see that while we are dealing with a delta potential, what this last expression implies is that neither 5p6x — » nor 8E8t —> for this system. Infinitesimal analysis of this system shows that 5£5/ a 6>5x « ft, (28) as expected. Conclusion The Lorentz invariant measure can be used as a conceptual foundation of quantum mechanics, and a direct link between special relativity and quantum mechanics can be presented without recourse to subsidiary conditions. In this paper, using infinitesimal analysis, a technique devel- oped at Cambridge beginning in Newton's time, we have derived the Heisenberg uncertainty principle from first principles. Acknowledgements Patrick would like to thank Dr. J. McKenna and Dr. M. McMillan for their valuable insights on the Lorentz invariant measure, and both Dr. F. Kaempffer and Dr. A. Zhitnisky for their unique insights on the Heisenberg uncertainty prin- ciple. References ] C. Moller. The Theory of Relativity, Oxford Uni- versity Press. 1972, p. 97. 2 J.L. Bell. A Primer of Infinitesimal Analysis. Cam- bridge University Press: Cambridge. 1998, p. 29. 3 R. Schlegel. Superposition and Interaction. Univer- sity of Chicago Press: Chicago. 1980, p. 198. l M. Jammer. The Conceptual Development of Quantum Mechanics. McGraw Hill: New York. 1966, p. 54. 5 P. Bruskiewich. "A Novel Look at the One Dimensional Delta Schrodinger Equation." Gamma Magazine. 138 (May 2005). Alicja Kielbik is a third year undergrad at the University of British Columbia (UBC). She is interested in quantum continuous measurement theory and its application to quantum computing and gravity. Patrick Bruskiewich is a UBC doctoral can- didate and editor-in-chief of CUPj. He can be reached at patrichh@phas.uhc.ca. * Canadian Undergraduate Physics Journal 1 7